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In Europe, important bridges have been built over the sea in recent years, such as fixed links across large stretches of water, for example, the Öresund Bridge 7.8 km long between Sweden

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www.engreferencebooks.com

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José J Oliveira Pedro

IST – University of Lisbon and GRID Consulting Engineers

Lisbon

Portugal

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Library of Congress Cataloging‐in‐Publication Data

Names: Reis, António J., 1949– author | Oliveira Pedro, José J., 1968– author.

Title: Bridge design : concepts and analysis / António J Reis, IST – University of Lisbon and Technical Director GRID Consulting Engineers, Lisbon, José J Oliveira Pedro, IST – University of Lisbon and GRID Consulting Engineers, Lisbon.

Subjects: LCSH: Bridges–Design and construction.

Classification: LCC TG300 (ebook) | LCC TG300 R45 2019 (print) | DDC 624.2/5–dc23

LC record available at https://lccn.loc.gov/2018041508

Cover Design: Wiley

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Set in 10/12pt Warnock by SPi Global, Pondicherry, India

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About the Authors xiii

1.5.4 Reinforced and Prestressed Concrete Bridges 24

1.5.5 Cable Supported Bridges 28

References 30

2 Bridge Design: Site Data and Basic Conditions 31

2.1 Design Phases and Methodology 31

2.2 Basic Site Data 32

2.3 Bridge Location Alignment, Bridge Length and Hydraulic Conditions 38

2.3.1 The Horizontal and Vertical Alignments 42

2.3.2 The Transverse Alignment 46

2.4 Elements Integrated in Bridge Decks 49

2.4.1 Road Bridges 49

2.4.1.1 Surfacing and Deck Waterproofing 50

2.4.1.2 Walkways, Parapets and Handrails 50

2.4.1.3 Fascia Beams 53

2.4.1.4 Drainage System 54

Contents

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2.4.1.5 Lighting System 55

2.4.1.6 Expansion Joints 55

2.4.2 Railway Decks 58

2.4.2.1 Track System 59

2.4.2.2 Power Traction System (Catenary System) 61

2.4.2.3 Footways, Parapets/Handrails, Drainage and Lighting Systems 61

References 61

3 Actions and Structural Safety 63

3.1 Types of Actions and Limit State Design 63

3.2 Permanent Actions 65

3.3 Highway Traffic Loading – Vertical Forces 68

3.4 Braking, Acceleration and Centrifugal Forces in Highway Bridges 72

3.5 Actions on Footways or Cycle Tracks and Parapets, of Highway Bridges 74

3.6 Actions for Abutments and Walls Adjacent

3.10.2 Dynamic Effects for Railway Bridges 82

3.11 Wind Actions and Aerodynamic Stability of Bridges 84

3.11.1 Design Wind Velocities and Peak Velocities Pressures 84

3.11.2 Wind as a Static Action on Bridge Decks and Piers 89

3.11.3 Aerodynamic Response: Basic Concepts 91

3.14 Shrinkage, Creep and Relaxation in Concrete Bridges 109

3.15 Actions Due to Imposed Deformations Differential Settlements 117

3.16 Actions Due to Friction in Bridge Bearings 119

3.17 Seismic Actions 119

3.17.1 Basis of Design 119

3.17.2 Response Spectrums for Bridge Seismic Analysis 121

3.18 Accidental Actions 124

3.19 Actions During Construction 124

3.20 Basic Criteria for Bridge Design 125

References 125

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4 Conceptual Design and Execution Methods 129

4.1 Concept Design: Introduction 129

4.2 Span Distribution and Deck Continuity 131

4.2.1 Span Layout 131

4.2.2 Deck Continuity and Expansion Joints 132

4.3 The Influence of the Execution Method 134

4.3.1 A Prestressed Concrete Box Girder Deck 134

4.3.2 A Steel‐Concrete Composite Steel Deck 136

4.3.3 Concept Design and Execution: Preliminary Conclusions 136

4.4 Superstructure: Concrete Bridges 138

4.4.1 Options for the Bridge Deck 138

4.4.2 The Concrete Material – Main Proprieties 139

4.4.2.1 Concrete 139

4.4.2.2 Reinforcing Steel 140

4.4.2.3 Prestressing Steel 140

4.4.3 Slab and Voided Slab Decks 142

4.4.4 Ribbed Slab and Slab‐Girder Decks 144

4.4.5 Precasted Slab‐Girder Decks 152

4.4.6 Box Girder Decks 155

4.5 Superstructure: Steel and Steel‐Concrete Composite Bridges 160

4.5.1 Options for Bridge Type: Plated Structures 160

4.5.2 Steels for Metal Bridges and Corrosion Protection 166

4.5.2.1 Materials and Weldability 166

4.5.2.2 Corrosion Protection 172

4.5.3 Slab Deck: Concrete Slabs and Orthotropic Plates 173

4.5.3.1 Concrete Slab Decks 174

4.5.3.2 Steel Orthotropic Plate Decks 176

4.5.4 Plate Girder Bridges 179

4.5.4.1 Superstructure Components 179

4.5.4.2 Preliminary Design of the Main Girders 182

4.5.4.3 Vertical Bracing System 188

4.5.4.4 Horizontal Bracing System 191

4.5.5 Box Girder Bridges 192

4.5.5.1 General 192

4.5.5.2 Superstructure Components 193

4.5.5.3 Pre‐Design of Composite Box Girder Sections 196

4.5.5.4 Pre‐Design of Diaphragms or Cross Frames 199

4.5.6 Typical Steel Quantities 201

4.6 Superstructure: Execution Methods 202

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4.6.2.7 Other Methods 222

4.6.3 Erection Methods for Steel and Composite Bridges 223

4.6.3.1 Erection Methods, Transport and Erection Joints 223

4.6.3.2 Erection with Cranes Supported from the Ground 224

4.7.2.3 Execution Method of the Deck and Pier Concept Design 233

4.7.2.4 Construction Methods for Piers 240

4.7.4.4 Special Bridge Foundations 247

4.7.4.5 Bridge Pier Foundations in Rivers 250

6.3.2 Overall Bending: Shear Lag Effects 283

6.3.3 Local Bending Effects: Influence Surfaces 287

6.3.4 Elastic Restraint of Deck Slabs 295

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6.3.5 Transverse Prestressing of Deck Slabs 297

6.3.6 Steel Orthotropic Plate Decks 300

6.4 Transverse Analysis of Bridge Decks 301

6.4.1 Use of Influence Lines for Transverse Load Distribution 301

6.4.2 Transverse Load Distribution Coefficients for Load Effects 302

6.4.3 Transverse Load Distribution Methods 303

6.4.3.1 Rigid Cross Beam Methods: Courbon Method 304

6.4.3.2 Transverse Load Distribution on Cross Beams 307

6.4.3.3 Extensions of the Courbon Method: Influence of Torsional Stiffness

of Main Girders and Deformability of Cross Beams 307

6.4.3.4 The Orthotropic Plate Approach 308

6.4.3.5 Other Transverse Load Distribution Methods 313

6.5 Deck Analysis by Grid and FEM Models 313

6.5.1 Grid Models 313

6.5.1.1 Fundamentals 313

6.5.1.2 Deck Modelling 315

6.5.1.3 Properties of Beam Elements in Grid Models 317

6.5.1.4 Limitations and Extensions of Plane Grid Modelling 318

6.5.2 FEM Models 318

6.5.2.1 Fundamentals 318

6.5.2.2 FEM for Analysis of Bridge Decks 323

6.6 Longitudinal Analysis of the Superstructure 329

6.6.1 Generalities – Geometrical Non‐Linear Effects: Cables and Arches 329

6.6.2 Frame and Arch Effects 332

6.6.3 Effect of Longitudinal Variation of Cross Sections 334

6.6.4 Torsion Effects in Bridge Decks – Non‐Uniform Torsion 336

6.6.5 Torsion in Steel‐Concrete Composite Decks 343

6.6.5.1 Composite Box Girder Decks 343

6.6.5.2 Composite Plate Girder Decks 345

6.6.5.3 Transverse Load Distribution in Open Section Decks 348

6.6.6 Curved Bridges 350

6.6.6.1 Statics of Curved Bridges 350

6.6.6.2 Simply Supported Curved Bridge Deck 352

6.6.6.3 Approximate Method 353

6.6.6.4 Bearing System and Deck Elongations 353

6.7 Influence of Construction Methods on Superstructure Analysis 355

6.7.1 Span by Span Erection of Prestressed Concrete Decks 356

6.7.2 Cantilever Construction of Prestressed Concrete Decks 357

6.7.3 Prestressed Concrete Decks with Prefabricated Girders 360

6.7.4 Steel‐Concrete Composite Decks 361

6.8 Prestressed Concrete Decks: Design Aspects 364

6.8.1 Generalities 364

6.8.2 Design Concepts and Basic Criteria 364

6.8.3 Durability 364

6.8.4 Concept of Partial Prestressed Concrete (PPC) 364

6.8.5 Particular Aspects of Bridges Built by Cantilevering 365

6.8.6 Ductility and Precasted Segmental Construction 366

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6.8.6.1 Internal and External Prestressing 367

6.8.7 Hyperstatic Prestressing Effects 367

6.8.8 Deflections, Vibration and Fatigue 368

6.9 Steel and Composite Decks 373

6.9.2 Design Criteria for ULS 373

6.9.3 Design Criteria for SLS 375

6.9.3.1 Stress Limitations and Web Breathing 376

6.9.3.2 Deflection Limitations and Vibrations 377

6.9.4 Design Criteria for Fatigue Limit State 377

6.9.5 Web Design of Plate and Box Girder Sections 383

6.9.5.1 Web Under in Plane Bending and Shear Forces 383

6.9.5.2 Flange Induced Buckling 385

6.9.5.3 Webs Under Patch Loading 387

6.9.5.4 Webs under Interaction of Internal Forces 389

6.9.6 Transverse Web Stiffeners 390

6.9.7 Stiffened Panels in Webs and Flanges 391

6.9.8 Diaphragms 394

6.10 Reference to Special Bridges: Bowstring Arches and Cable‐Stayed

Bridges 395

6.10.2 Bowstring Arch Bridges 396

6.10.2.1 Geometry, Slenderness and Stability 396

6.10.2.2 Hanger System and Anchorages 402

6.10.2.3 Analysis of the Superstructure 403

6.10.3 Cable‐Stayed Bridges 404

6.10.3.1 Basic Concepts 404

6.10.3.2 Total and Partial Adjustment Staying Options 408

6.10.3.3 Deck Slenderness, Static and Aerodynamic Stability 411

6.10.3.4 Stays and Stay Cable Anchorages 414

6.10.3.5 Analysis of the Superstructure 416

References 418

7 Substructure: Analysis and Design 423

7.1 Introduction 423

7.2 Distribution of Forces Between Piers and Abutments 423

7.2.1 Distribution of a Longitudinal Force 423

7.2.2 Action Due to Imposed Deformations 424

7.2.3 Distribution of a Transverse Horizontal Force 425

7.2.4 Effect of Deformation of Bearings and Foundations 429

7.3 Design of Bridge Bearings 430

7.3.1 Bearing Types 430

7.3.2 Elastomeric Bearings 430

7.3.3 Neoprene‐Teflon Bridge Bearings 434

7.3.4 Elastomeric ‘Pot Bearings’ 435

7.3.5 Metal Bearings 437

7.3.6 Concrete Hinges 439

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7.4 Reference to Seismic Devices 441

7.4.1 Concept 441

7.4.2 Seismic Dampers 441

7.5 Abutments: Analysis and Design 444

7.5.1 Actions and Design Criteria 444

7.5.2 Front and Wing Walls 446

7.5.3 Anchored Abutments 448

7.6 Bridge Piers: Analysis and Design 449

7.6.1 Basic Concepts 449

7.6.1.1 Pre‐design 449

7.6.1.2 Slenderness and Elastic Critical Load 449

7.6.1.3 The Effect of Geometrical Initial Imperfections 450

7.6.1.4 The Effect of Cracking in Concrete Bridge Piers 450

7.6.1.5 Bridge Piers as ‘Beam Columns’ 451

7.6.1.6 The Effect of Imposed Displacements 452

7.6.1.7 The Overall Stability of a Bridge Structure 453

7.6.1.8 Design Bucking Length of Bridge Piers 453

7.6.2 Elastic Analysis of Bridge Piers 454

7.6.3 Elastoplastic Analysis of Bridge Piers: Ultimate Resistance 459

7.6.4 Creep Effects on Concrete Bridge Piers 465

7.6.5 Analysis of Bridge Piers by Numerical Methods 465

7.6.6 Overall Stability of a Bridge Structure 471

References 473

8 Design Examples: Concrete and Composite Options 475

8.1 Introduction 475

8.2 Basic Data and Bridge Options 475

8.2.1 Bridge Function and Layout 475

8.2.2 Typical Deck Cross Sections 476

8.2.3 Piers, Abutments and Foundations 477

8.2.4 Materials Adopted 477

8.2.4.1 Prestressed Concrete Deck 478

8.2.4.2 Steel‐concrete Composite Deck 481

8.2.5 Deck Construction 481

8.3 Hazard Scenarios and Actions 481

8.3.1 Limit States and Structural Safety 482

8.3.2 Actions 482

8.3.2.1 Permanent Actions and Imposed Deformations 482

8.3.2.2 Variable Actions 484

8.4 Prestressed Concrete Solution 486

8.4.1 Preliminary Design of the Deck 486

8.4.2 Structural Analysis and Slab Checks 486

8.4.3 Structural Analysis of the Main Girders 492

8.4.3.1 Traffic Loads: Transverse and Longitudinal Locations 493

8.4.3.2 Internal Forces 497

8.4.3.3 Prestressing Layout and Hyperstatic Effects 497

8.4.3.4 Influence of the Construction Stages 498

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8.4.4 Structural Safety Checks: Longitudinal Direction 498

8.4.4.1 Decompression Limit State – Prestressing Design 498

8.4.4.2 Ultimate Limit States – Bending and Shear Resistance 501

8.5 Steel–Concrete Composite Solution 502

8.5.1 Preliminary Design of the Deck 502

8.5.2 Structural Analysis and Slab Design Checks 503

8.5.3 Structural Analysis of the Main Girders 503

8.5.3.1 Traffic Loads Transverse and Longitudinal Positioning 504

8.5.3.2 Internal Forces 505

8.5.3.3 Shrinkage Effects 505

8.5.3.4 Imposed Deformation Effect 506

8.5.3.5 Influence of the Construction Stages 506

8.5.4 Safety Checks: Longitudinal Direction 507

8.5.4.1 Ultimate Limit States – Bending and Shear Resistance 507

8.5.4.2 Serviceability Limit States – Stresses and Crack Widths Control 509

References 510

Annex A: Buckling and Ultimate Strength of Flat Plates 511

A.1 Critical Stresses and Buckling Modes of Flat Plates 511

A.1.1 Plate Simply Supported along the four Edges and under

a Uniform Compression (ψ = 1) 511

A.1.2 Bending of Long Rectangular Plates Supported at both Longitudinal Edges or

with a Free Edge 513

A.1.3 Buckling of Rectangular Plates under Shear 513

A.2 Buckling of Stiffened Plates 514

A.2.1 Plates with One Longitudinal Stiffener at the Centreline under Uniform

Compression 515

A.2.2 Plate with Two Stiffeners under Uniform Compression 516

A.2.3 Plates with Three or More Longitudinal Stiffeners 517

A.2.4 Stiffened Plates under Variable Compression Approximate Formulas 518

A.3 Post‐Buckling Behaviour and Ultimate Strength of Flat Plates 518

A.3.1 Effective Width Concept 519

A.3.2 Effective Width Formulas 520

References 523

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António J Reis became a Civil Engineer at IST – University of Lisbon in 1972 and

obtained his Ph.D at the University of Waterloo in Canada in 1977 He was Science Research Fellow at the University of Surrey, UK, and Professor of Bridges and Structural Engineering at the University of Lisbon for more than 35 years Reis was also Visiting Professor at EPFL Lausanne Switzerland in 2013 and 2015 In 1980, he established his own design office GRID where he is currently Technical Director and was responsible for the design of more than 200 bridges The academic and design experience were always combined in developing and supervising research studies and innovative design aspects in the field of steel and concrete bridges, cable stayed bridges, long span roofs and stability of steel structures A Reis has design studies and projects in more than

20 countries, namely in Europe, Middle East and Africa and presented more than 150 publications He received several awards at international level from IABSE, ECCS, ICE and Royal Academy of Sciences of Belgium

José J Oliveira Pedro became a Civil Engineer at IST – University of Lisbon in 1991,

concluding his Master’s degree in 1995 and Ph.D in 2007, with the thesis “Structural analysis of composite steel-concrete cable-stayed bridges” He joined the Civil Engineering Department of IST in 1990, as a Student Lecturer, and is currently Assistant Professor of Bridges, Design of Structures and Special Structures In 1999, he was Researcher at Liège University / Bureau d’Etudes Greisch and, in 2015, Visiting Professor

at EPFL Lausanne In 1991, he joined design office GRID Consulting Engineers, and since then is very much involved in the structural design of bridges and viaducts, stadi-ums, long span halls and other large structures He is the author/co-author of over seventy publications in scientific journals and conference proceedings In 2013, he received the Baker medal, and in 2017 the John Henry Garrood King Medal, from the

Institute of Civil Engineers, for the best papers published in Bridge Engineering journal.

About the Authors

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About 15 years ago, the first author, A J Reis, was invited by Wiley to write a book on

Bridge Design that could be adopted as textbook for bridge courses and as a guideline

for bridge engineers The author’s bridge course notes from the University of Lisbon, updated over almost 30 years, were the basis for this book For different reasons, the completion of the book was successively postponed until a final joint effort with the second author, J J Oliveira Pedro, made this long project a reality The book mainly reflects the long design experience of the authors and their academic lecturing and research activities

Bridge design is a multidisciplinary activity It requires a good knowledge and standing of a variety of aspects well beyond structural engineering Road and railway design, geotechnical and hydraulic engineering, urban planning or environmental impact and landscape integration are key aspects Architectural, aesthetic and environ-mental aspects are nowadays recognized as main engineering issues for bridge design-ers However, these subjects cannot be studied independently of structural and construction aspects, such as the bridge erection method On the other hand, what differentiates bridge design from building design, for example, is generally the role of the bridge engineer as a leader of the design process Hence, the first aim of this book is

under-to present an overview on all these aspects, discussing from the first bridge concepts under-to analysis in a unified approach to bridge design

The choice of structural materials and the options for a specific bridge type are part

of the design process Therefore, the second aim of the book is to discuss concepts and principles of bridge design for the most common cases – steel, concrete or composite bridges Good bridge concepts should be based on simple models, reflecting the struc-tural behaviour and justifying design options Sophisticated modelling nowadays adopts available software, most useful at advanced stages of the design process However, it should be borne in mind that complex modelling does not make necessarily a good bridge concept

The methodology to select the appropriate bridge typology and structural material is discussed in the first four chapters of this book Examples, mainly from the authors’ design experiences, are included General aspects and bridge design data are presented

in Chapter 2 Actions on bridges are included in Chapter 3 with reference to the Eurocodes Structural safety concepts for bridge structures and limit state design criteria are also outlined in this chapter Chapter 4 includes the conceptual design of bridge super‐ and substructures Basic concepts for prestressed concrete, steel or steel

Preface

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concrete composite bridges, with slab, slab‐girder and box girder decks are dealt with These topics are discussed in relation to superstructures and execution methods such

as classical falsework, formwork launching girders, incremental launching and balanced cantilevering Bridge substructures are referred to in Chapter 4 as well, namely for the basic typologies of bridge piers, abutments and foundations

Architectural, environmental, and aesthetic aspects that could be adopted as primary guidelines when developing a bridge concept are addressed in Chapter 5 Principles are explained on the basis of design cases from the authors’ design practices Of course, this could have been done on the basis of many other bridges However, it is sometimes difficult to comment on bridge aesthetics while not being aware of design, cost or execution constraints faced by other designers

Specific aspects of structural analysis and design are dealt with in Chapters 6 and 7 Particular reference is made in Chapter 6 to simplified approaches to the preliminary superstructure design These approaches can also be adopted to check results from sophisticated numerical models at the detailed design stages The influence of the erection method on structural analysis and design of prestressed concrete, steel and composite bridge superstructures is considered in Chapter 6 Particular reference is made to safety during construction stages and redistribution of internal forces due to time dependent effects Chapter 6 ends with some design concepts and analysis for bowstring arch bridges and cable-stayed bridges Of course, due to the scope of the book, the aspects dealt with for these specific bridge types are introductory in nature.The substructure structural analysis and design is presented in Chapter 7 The distri-bution of horizontal forces between piers and abutments due to thermal, wind and earthquake actions is discussed Stability of bridge piers and reinforced concrete design aspects are dealt with Bridge bearing typologies and specifications are introduced Particular reference is made to bridge seismic isolation and different types of seismic isolation devices are presented

The book ends with Chapter 8, which presents a simple design case with two different superstructure solutions – a prestressed concrete deck and a steel‐concrete composite deck The application of design principles presented throughout the book is outlined.The authors expect readers may find this book useful and in some way it will contrib-ute to bridges reflecting the ‘art of structural engineering’

António J Reis and José J Oliveira Pedro

Lisbon, May 2018

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This book is the result of the authors’ activities at IST–University of Lisbon and at GRID Consulting Engineers The support of both institutions is a pleasure to acknowledge and special thanks are due to Professor Francisco Virtuoso from IST and

to our colleagues from GRID

During 45 years of professional life as designer, the first author, A Reis, had the privilege of meeting a few outstanding bridge engineers Particular reference is made to Jean Marie Cremer, from Bureau d’Études Greisch, with whom A Reis had the pleasure

of working with on a few bridge projects but, most important, developing a ship with

friend-Part of this book was written by the first author, A Reis, during his stays in 2013 and

2015 as Visiting Professor at EPFL École Polytechnique Féderale de Lausanne, Switzerland The second author, J Pedro, had a similar opportunity in 2015 Thanks are due to EPFL and, in particular, to Professor Alain Nussbaumer for these opportunities.The authors are also grateful to all sources and organizations allowing the reproduc-tion of some figures and pictures with due credit referenced in the text

Last, but not least, thanks are due to our families for the time this book has taken from being with them

António J Reis and José J Oliveira Pedro

Lisbon, May 2018

Acknowledgements

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Bridge Design: Concepts and Analysis, First Edition António J Reis and José J Oliveira Pedro

1

1.1 Generalities

Bridges are one of the most attractive structures in the field of Civil Engineering, creating aesthetical judgements from society and deserving, in many cases, the Latin designation

in the French language of Ouvrages d’Art.

Firstly, a set of definitions and appropriated terminology related to bridge structures

is established before discussing bridge design concepts A short historical view of the topic is included in this chapter to introduce the reader to the bridge field, going from basic concepts and design methods to construction technology

A bridge cannot be designed without an appropriated knowledge of general concepts that go well beyond the field of structural analysis and design The concept for a bridge requires from the designer a general knowledge of other aspects, such as environmental and aesthetic concepts, urban planning, landscape integration, hydraulic and geotechnical engineering

The designer very often has to discuss specific problems for a bridge design concept with specialists in other fields, such as the ones previously mentioned, as well as from aspects of more closely related fields like highway or railway engineering

Introducing the reader to the relationships between all the fields related to bridge design, from the development of the bridge concept to more specific aspects of bridge construction methods, is one of the aims of this book

Most of the bridge examples are based on design projects developed at the author’s design office Some of these design cases have been summarized in the chapters in order to illustrate the basic concepts developed throughout the book

1.2 Definitions and Terminology

A bridge may be defined as a structure to traverse an obstacle, namely a river, a valley, a

roadway or a railway The general term bridge is very often left for the first case, that is,

a structure over a river leaving the more specific term of viaduct for bridges over valleys

or over other obstacles So, the relevance of the structure very often related to its length

or main span has nothing to do with the use of the terms bridge or viaduct One may have bridges of only 20 m length and viaducts 3 or 4 km long In highway bridge terminology, it is usual to differentiate between viaducts passing over or under a main

Introduction

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road by designating them as overpasses or underpasses So, one shall adopt the term

‘bridge’ to designate bridges in particular, or viaducts Figure 1.1 shows a bridge over the river Douro that is 703 m long, 36 m width for eight traffic lanes and has a main span of

150 m, and also a viaduct in Madeira Island, 600 m long for four traffic lanes and with a typical span of 45 m The decks of these structures are made of two parallel box girders supported by independent piers

(a)

(b)

Figure 1.1 (a) The Freixo Bridge over the river Douro in Oporto, 1993, and (b) a viaduct in Madeira

Island, Portugal, 1997 (Source: Courtesy GRID, SA).

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In Europe, important bridges have been built over the sea in recent years, such as fixed links across large stretches of water, for example, the Öresund Bridge (7.8 km long) between Sweden and Denmark, and the Vasco da Gama Bridge (12 km long) in the Tagus river estuary, Lisbon, shown in Figure 1.2.

Many of these structures include main spans as part of cable‐stayed or sion bridges and many typical spans repeated along offshore or inland areas If that occurs over the riversides, it is usual to designate that part of the bridge the

1998 (Source: Photograph by José Araujo).

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A bridge integrates two main parts:

● the superstructure; the part traversing the obstacle and

● the substructure; the part supporting the superstructure and transferring its loads to

the ground through the foundations

The superstructure is basically made of a deck transferring the loads to the piers by bearings or by a rigid connection between the deck and the pier; the substructure includes the piers, foundations and the abutments, as shown in Figure 1.3 The piers transfer the loads from the superstructure due to permanent and variable actions, namely dead weight, traffic loads, thermal, wind and earthquake action, to the foundations The abutments establish the transition between the superstructure and the earthfill of the highway or the railway and retain the filling material The abutments transfer the loads induced by the superstructure, generally transmitted by the bearings, and supporting the soil impulses generated by the embankments

The deck is, in general, supported by a set of bearings, some located at the abutments,

as previously referred to, and others located at the top of the piers as shown in Figure 1.3 Nowadays, these bearings are generally made of elastomeric materials (natural rubber

or synthetic rubber – chloroprene) and steel

The foundations of the bridge piers and abutments may be by footings, as in Figure 1.3 (shallow foundations) or by piles (deep foundations) A different type of foundation include caissons made by lowering precasted segmental elements in a previous exca-vated soil, a method adopted sometimes for deep bridge piers foundations in rivers

1.3 Bridge Classification

Bridges may be classified according several criteria namely:

● the bridge function, dependent on the type of use of the bridge, giving rise to

designa-tions of highway or railway bridges, canal bridges for the transportation of water, quay bridges in ports, runway or taxiway bridges in airports, pedestrian bridges or pipeline bridges The function of the bridge may be twofold as for example in the case

of the Oresund Bridge, for railway and highway traffic (Figure 1.2)

● the bridge structural material, like masonry bridges, as used in the old days since the

Romans, timber bridges, metal bridges in steel or aluminium or in iron as adopted in the nineteenth century, concrete bridges either in reinforced concrete or prestressed

concrete (more precisely, partially prestressed concrete as preferred nowadays) and,

more recently, composite steel‐concrete bridges

● the bridge structural system, which may be distinguished by:

– the longitudinal structural system;

– the transversal structural system

The former, the longitudinal system, gives rise to beam bridges, frame bridges, arch bridges and cable supported bridges; namely, cable‐stayed bridges and suspen-sion bridges The last, the transversal structural system, is characterized by the type adopted for the cross section of the superstructure, namely slab, girder or box girder bridges A preliminary discussion on bridge structural systems is presented in next section

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35.00 35.00 35.00 35.00 35.00 30.00 35.00

ORIGINAL EARTHFILL 104.0

L C

Figure 1.3 Section elevation and typical cross section of a bridge – The Lugela bridge in Mozambique, 2008 Superstructure (deck) and Substructure

(piers, abutments and foundations).

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● Another type of classification is often adopted, according to:

– the predicted lifetime of the bridge, namely temporary (made in general in wood or steel) or definitive bridges

– the fixity of the bridge, namely fixed or movable bridges, like lift bridges if the deck may be raised vertically rolling bridges if the deck rolls longitudinally, or swing bridges if the deck rotates around a vertical axis

– the in‐plan geometry of the bridge, like straight, skew or curved bridges

1.4 Bridge Typology

Different bridge typologies, namely concerning the longitudinal structural system or the deck cross section, may be adopted with different structural materials The concept design of a bridge is developed mainly in Chapters 4 and 5, but a brief description of the variety of bridge options is presented here in order to introduce the topics of Chapters 2 and 3 concerning the basic data and conditions for design

Nowadays, a beam bridge is the most usual type where the deck is a simple slab, a beam and slab (Figure 1.3) or a box girder deck Beam bridges may be adopted in reinforced

concrete for small spans (l), generally up to 20 m, or in prestressed concrete or in steel‐

concrete composite decks (Figure 1.4) for spans up to 200 m or even more The structure may have a single span, simply supported at the abutments, or multiple continuous spans (Figure 1.5) Between these two cases, some other bridge solutions are possible like multiple span decks, in which most of the spans are continuous, but some spans have internal hinges like in the so called ‘Gerber’ type beam bridges, shown in Figure 1.6 However, the general trend nowadays is to adopt, as far as possible, fully con-tinuous superstructures, to reduce maintenance of the expansion joints and to improve the earthquake resistance of the bridge if located in a seismic region Continuous decks more than 2000 m long have been adopted for beam bridges, either for road or rail bridges Yet, in long continuous bridges, the distance between expansion joints is generally restricted to 300–600 m to reduce displacements at the expansion joints In a beam bridge, the connection between the superstructure and the piers is made by bearings, as in Figure 1.3, which allow the relative rotations between the deck and the piers; the relative longitudinal displacements between the deck and the piers may or may not be restricted, depending on the flexibility and slenderness of the piers, as discussed in Chapters 4 and 7

super-If the deck is rigidly connected to the piers, one has a frame bridge (Figure 1.7) The superstructure may be rigidly connected to some piers and standing in some bearings, allowing rotations, or rotations and displacements, between the deck and some of other piers

In frame bridges, the piers are in most cases vertical However, frame bridges with slant legs, exemplified in Figure 1.8a, are a possible option For a frame bridge with slant legs or arch bridges, the main condition for adopting these typologies is the load bear-ing capacity of the slopes of the valley to accommodate, with very small displacements, the horizontal component H (the thrust of the arch) of the force reactions induced by the structure, as shown in Figure 1.8b

An arch is likely to be a very efficient type of structure, an aesthetically pleasant tion for long spans in deep valleys, provided the geological conditions are appropriate The ideal shape of the arch, if the load transferred from the deck is considered as a www.engreferencebooks.com

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solu-uniformly distributed load q (valid for closed posts), is a second degree parabola because

the arch for the permanent load is free from bending moments In this case, the arch is only subjected to axial forces; that is, the arch follows the ‘pressure line’ It is easy to show using simple static equilibrium (bending moment condition equal to zero at the

crown) that the thrust is given by H ql2/ 8f

Arch bridges may have different typologies and be made of different structural rials In the old days, masonry arches made of stones were very often adopted for small

mate-to medium span bridges More recently, iron, steel and reinforced concrete bridges replaced these solutions with spans going up to several hundred metres One of the

Figure 1.4 Steel‐concrete composite plate girder decks: Approach viaducts three‐dimensional model

of the Sado River railway crossing, in Portugal (Figure 1.12).

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most beautiful arch bridges is Arrábida Bridge, in Oporto (Figure 1.9), designed at the end of the 1950s, the beginning of the 1960s and opened to traffic in 1963 The bridge,

at the time the longest reinforced concrete arch bridge in the world, has a span of 270 m

and a rise of 54 m ( f/l = 1/5).

The arch bridge may have the deck working from above or from below, as shown in Figures 1.10 and 1.11 This last solution is adopted for traversing rivers at low levels

above the water, with particular restrictions for the vertical clearance h for navigation

channels The horizontal component of the reaction at the base of the arch, at the

con-nection between the arch and the deck, is taken by the deck A bowstring arch bridge is

the designation for this bridge type, in which the deck has a tie effect, together with its beam behaviour Figure 1.12 shows a multiple bowstring arch bridge, with a continuous deck composed of a single steel box section The deck, with spans of 160 m, is a steel‐concrete composite box girder to allow the required torsion resistance under eccentric traffic loading However, the classical solution for bowstring arches is made of a beam and slab deck suspended from above by two vertical or inclined arches, as presented in Chapter 6

The main restriction nowadays for the construction of arches is the difficulty of the execution method, when compared to a long span frame bridge with vertical piers, built

by the balanced cantilever method referred to in Chapter 4

For spans above 150 m and up to 1000 m, cable‐stayed bridges, as previously shown in Figure 1.2, are nowadays generally preferred to beam or frame bridges, for which the

H H

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crown post

Figure 1.10 Arch bridges: (a) the classical parabolic two hinges arch bridge; (b) structural longitudinal

model; (c) independent arch and deck at the crown; (d) segmental arch and (e) low rise arch for a pedestrian bridge without posts.

Figure 1.9 The Arrabida Bridge in Oporto, Portugal, 1963 (Source: Photograph by Joseolgon / https://

commons.wikimedia.org / Public Domain).

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Figure 1.12 A bowstring arch for a railway bridge: (a) the crossing of the Sado River in Alcacer do Sal,

Portugal, 2010 and (b) deck cross section – a steel concrete composite box girder (Source: Courtesy

GRID, SA).

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longest span is about 300 m Even for spans bellow 100 m, cable‐stayed bridges have been adopted as structurally efficient and aesthetically pleasant solutions; for example,

in urban spaces where very slender decks are required The basic schemes for the stay arrangement in cable‐stayed bridges are shown in Figure 1.13 – the fan, semi‐fan or harp arrangement The semi‐fan arrangement is the most adopted one for economy of stay cable quantity For aesthetics, usually the harp arrangement is the preferred one, since it reduces the visual impact of crossing cables for skew views of the bridge, as is apparent from Figure 1.2

In cable‐stayed bridges with spans up to 500 m the deck may be made of concrete, but above this span length, steel or steel‐concrete composite decks are preferred, to reduce the dead weight of the superstructure The cable‐stayed bridge deck is subjected to large compressive forces induced by the stay cables, as shown in Figure 1.14 In the first generation of cable‐stayed bridges, the decks where in steel and the stay‐cable anchor-ages were kept at a considerable distance at the deck level In these bridges, the beam load effect in the deck was relevant In the last few decades, a new generation of cable‐stayed bridges has been developed with multiple stay cables anchored at small distances

at the deck level, very often between 5 and 15 m, allowing a considerable reduction of the beam internal forces in the deck This is the case for cable‐stayed bridges with very slender concrete decks with a span/height relationship for the deck that can reach very high values up to 400

In cable‐stayed bridges, the deck is supported by a single plan or two plans of stay

cables The former suspension scheme is designated axial or central suspension while the last one is called lateral suspension The axial suspension scheme requires a deck

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with considerable torsion rigidity, generally a box girder deck, to support the asymmetric traffic loadings as well as to improve its aerodynamic stability Figure 1.15 shows one of the most remarkable bridges built in 1977, at the time a world record for cable‐stayed bridges with a prestressed concrete deck On the contrary, in lateral suspension cable‐stayed bridges, the deck may be reduced to a simple slab supported laterally by stay cables, or to a slab and girder section with an open configuration, since torsion resistance is assured by the staying scheme, as is the case for the Vasco da Gama Bridge (Figure 1.2b).

The other type of cable supported bridge is, as previously referred to, the suspension bridge A suspension bridge includes a stiffening girder, the main cables, towers, hang-ers and anchoring blocks, as shown in Figure 1.16

The main cables are usually externally anchored (earth anchored cables), but in some

bridges with smaller spans it is possible to adopt a self‐anchored suspension bridge by

anchoring the cables at the deck, as shown in Figure 1.16 In the former, the tension forces are transferred directly to the ground while, in the latter, the cable forces are transferred (as large compression forces) to the deck

The stiffening girder of a suspension bridge may be made of two parallel trusses,

as the more classical suspension bridges developed in the North America during the twentieth century A second generation of suspension bridges was introduced

in Europe in the second part of the twentieth century, by replacing the ening girder by a streamline steel box girder deck and diagonal hanger ropes greatly improving the aerodynamic stability of the deck In Figure 1.17, two of these bridge stiffening girder typologies are shown – the Akashi Kaikyō Bridge completed in 1998 in Japan, presently the world’s longest span at 1991 m, and the Great Belt East Bridge, also completed in 1998 but in Denmark, with a central span of 1624 m

truss‐stiff-In a suspension bridge, the permanent loads of the deck are taken by the cable system The main cables adopt a parabolic configuration under the uniform dead load transmit-ted from the deck through the hangers The live loads are basically taken by combined local bending action of the deck, between the hangers, and the main cable

QNT

Stay-cables

Deck Tower

Figure 1.14 Cable‐stayed bridges: static equilibrium at deck level with axial forces induced by

the stays.

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Figure 1.15 An example of a cable‐stayed bridge with axial suspension: (a) the Brotonne Bridge,

France, 1977, with a main span of 320 m (Source: Photograph by Francis Cormon) with (b) a concrete

box girder deck cross section.

Hangers Main cable

(b)

(a)

Tower

Cable anchorage

Sag Deck Tower sanddle

Figure 1.16 (a) Basic layout and notation of suspension bridges (b) Externally anchored and

self‐anchored suspension bridges.

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35.50 m

27.00 m 23.60

31.00 4.00

Figure 1.17 (a) Akashi Kaikyō Bridge, Japan, 1998, with a truss‐stiffening girder deck (Source: Photograph by Pinqui/ https://commons.wikimedia.org) and

(b) Great Belt Bridge, Denmark, 1998, with a streamline box girder deck (Source: Photograph by Tone V V Rosbach Jensen/ https://commons.wikimedia.org).

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1.5 Some Historical References

The history of bridges is well‐documented in a variety of excellent references [1–4] To keep this section within the scope envisaged for this book, a short historical review will present the roles of some architects, engineers and bridge builders

Materials and structural shapes have always been the key elements to understand how bridges were conceived and built throughout the centuries For bridge structural materials, one may considerer different stages grouped as follows:

● Stone/masonry and wood bridges;

● Metal–iron and steel bridges;

● Concrete reinforced concrete and prestressed concrete bridges

The first group, stone/masonry and wood bridges, includes most of the bridge history from the Romans to the eighteenth century, while the second and third groups (metal and concrete bridges) may be included in bridge history from the eighteenth century to the present

Structural shapes and static schemes were also key factors in bridge historical development that, in short, may be summarized as:

● Arch bridges;

● Beam, frame and truss bridges;

● Cable supported bridges – suspension and cable‐stayed bridges

Contrary to what happens with bridge structural materials, it is not possible to include these structural typologies in different ages along bridge history Even for cable supported bridges, one may find references to the use of ropes made with natural fibres

to achieve a resistant structure in the early ages, even before the Roman Bridges Nowadays, arches, introduced by the Romans, are made with different materials (concrete or steel), still reflecting the art of structural engineering

1.5.1 Masonry Bridges

Apart from the earliest records of bridges, the first appears to be 600 bce, engineering bridge history may be considered to have been initiated by the Romans The Romans were the introducers of science in arch construction providing many examples of mag-nificent masonry bridges and aqueducts such as the ‘Pont du Gard’ in France and

‘Puente de Alcantara’ (Figure 1.18) over the river Tagus, Spain, from the second century [4] The former is an aqueduct with a total length of 275 m, a maximum depth of 49 m and spans 22 m The last is a masonry bridge as well, but with spans reaching 28 m and piles reaching 47 m high, appointed the most significant achievement of Roman engi-neering Roman bridges are based on the concept of semi‐circular arches, transferring the thrust to piers, and large piers require about one‐third of the spans in multiple arch bridges The Romans also introduced lime mortar and pozollanic cement for realizing voussoir arches, contributing to increasing span lengths and durability

Up until the end of the eighteenth century, bridges were built with masonry or wood Some beautiful examples built along the centuries may still be appreciated, such as Santa Trinita Bridge in Florence (Figure 1.19) built with masonry in the sixteenth century and www.engreferencebooks.com

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Figure 1.19 Santa Trinita Bridge in Florence, sixteenth century (Source: Photography by Bruno Barral/

https://commons.wikimedia.org).

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designed by the architect Bartolomeo Ammanati While the classical Roman Bridges were made with circular arches, the elegance of Santa Trinita Bridge is due to three flattened elliptic arches, which are the oldest in the world At the beginning of the twentieth century,

in 1905 to be precise, the span record for masonry bridges was reached with the tion of the Plauen Bridge [1, 3], in Germany, which has a main span of 90 m

construc-1.5.2 Timber Bridges

In the eighteenth century, some remarkable wood bridges were built In Switzerland, due to

a long tradition of wood construction builders, remarkable structures were built, like the Shaffhouse Bridge, designed in 1775 by Grubleman, with two continuous spans of 52 and

59 m In the USA, one of the most famous wood bridges was built in 1812, over the river Schuylkill with a main span of 104 m, unfortunately destroyed by a fire some years after its construction Most old timber bridges have disappeared along history; an exception, pres-ently the oldest wooden bridge to our knowledge, is the Kappellbruke (Chapel Bridge) erected originally in 1333 over Lake Lucerne, Switzerland (Figure 1.20) This footbridge was also destroyed by a fire in 1993 and completely rebuilt, standing now as a world heritage site

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materials From this time on, metal truss bridges and suspension bridges were adopted with increasing span lengths.

The first metallic bridge built in the world may be considered to be the Coalbrookdale Bridge [4] with a span of 30 m, designed by Abraham Darby III, a cast iron structure built

in Great Britain in 1779 But development had to wait until the end of the nineteenth century, when the prices for steel production has dropped down, to realize the first main steel bridges One of the most decisive contributions for the development of metal bridges was from Thomas Telford in the nineteenth century Telford was an English engineer with

a large number of relevant projects, always taking aesthetics as a key issue for his work, in parallel with the development of new structural and construction schemes Craigelachie Bridge, in Scotland, shown in Figure 1.21, is an excellent example of the aesthetical rele-vance of Telford’s design work In 1826, a famous eye bar wrought iron chain suspension bridge (Figure 1.22), with a main span of 176 m, also from Thomas Telford, was completed

in the UK over the Menai Strait, achieving the world record for span length The iron chain was replaced in 1938 by pin steel bars, allowing the bridge to remain in service up until now The construction of long span beam bridges is considered to have been initi-ated with the first box girder bridge in wrought iron, the Britannia Bridge in Wales, UK, built in 1850 by Robert Stephenson (Figure 1.23) The Britannia Bridge, with two main spans of 146 m, had a rectangular cross section with a railroad inside After a fire in 1970, this famous bridge was modified in 1971 by inserting a truss arch underneath and

Figure 1.21 Craigellachie Bridge, 45 m span, over the Spey River, Scotland, 1815 (Source: Photograph

by Craig Williams/ https://commons.wikimedia.org).

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Figure 1.23 The Britannia Bridge, 1850, North Wales (Source: Courtesy of the Los Angeles County

Museum of Art).

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carrying road and rail traffic, but seriously affecting the architecture of the bridge The type of box girder solution adopted for the Britannia Bridge was not retained due to the development of truss arch bridges at the end of the nineteenth century.

Reductions in steel prices allowed the increased development of steel arch bridges, or more precisely, wrought iron, with particular reference to long span bridges in the USA The most famous one was the San Luis Bridge, over the Mississippi, concluded in 1874 with three arches of 152 + 157 + 152 m, and in Europe with several famous arch bridges from the Gustave Eiffel Society The first relevant Eiffel’s bridges were the Maria Pia Bridge in Oporto, Portugal [5], in 1877, and the Viaduc du Gabarit, in France, in 1884 The former (Figure 1.24) 563 m long and with a main span of 160 m, has a truss arch for the railway deck of only 6 m Particular reference should be made to the erection scheme

of the arch using cantilever construction The Viaduct du Gabarit has a main span arch

of 165 m, 52 m rise and a total length of 564 m This arch was also erected by cantilever

as per the Maria Pia Bridge In 1885, the Belgium engineer Théophile Seyrig, who had worked with Eiffel, designed the Luiz I Bridge [6], also in Oporto and very close to the Maria Pia Bridge The Luiz I Bridge, an arch bridge, was the largest span in the world at its time – 174.5 m (Figure 1.25a) This bridge with two road decks has a two hinged arch

at the base, in spite of its appearance, due to the need of inserting the lower deck The bridge is located in a classified UNESCO World Heritage Site A great deal of strength-ening of the bridge was done in 2005 (Figure 1.25b), including its arch, truss girder and piers, with an entire replacement of the upper deck with a new steel deck, adapting it for the Metro of Oporto trains (Figure 1.26) The reduction in dead weight of the existing deck, which included concrete and brick elements inserted many years after the original deck, was achieved The reductions in deck dead load allowed a significant reduction in the strengthening of the arch required for the new functional conditions and required

Figure 1.24 Maria Pia Bridge, 1877, Oporto, Portugal (Source: Photograph by José O Pedro).

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strengthening of the bridge The design of the upgrading of the bridge was made out affecting its appearance as required by the design and build conditions of the bid.With the development of industrial production of steel, after the introduction by Henry Bessemer in 1856 of its patent, the Bessemer Converter, transforming cast iron

with-in to the much better steel material, with-in terms of tensile resistance and ductility, allowed the building of long span trusses in the United States and Europe The first long span European bridge was the Firth of Forth Bridge (Figure 1.27) built between 1881 and

1890, with two main spans of 521 m and two side spans of 207 m The two main spans were made with two cantilevers of 207 m supporting a central part made of truss beam

107 m in length The development of these types of bridges was not unfortunately made

(a)

(b)

Figure 1.25 (a) Luiz I Bridge, 1885, Oporto, Portugal: (a) before and (b) during the upgrading works

(Source: Courtesy GRID, SA).

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without the occurrence of some historical accidents, such as the one occurring in the railway bridge over the river Tay, Scotland, in 1879 and the accident during the erection

of the Quebec Bridge, Canada, in 1907 The Quebec Bridge with spans of 549 m was concluded only after a second accident during construction in 1917

Figure 1.26 Luiz I Bridge after being upgraded (Source: Photograph by José Araujo).

Figure 1.27 Firth of Forth Bridge, 1890, Scotland (Source: Photograph by Andrew Shiva/Godot13,

https://commons.wikimedia.org/wiki/File:Scotland‐2016‐Aerial‐Edinburgh‐Forth_Bridge.jpg CC

BY‐SA 4.0).

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1.5.4 Reinforced and Prestressed Concrete Bridges

The lack of sufficient tensile resistance makes concrete inadequate as a bridge

struc-tural material However, concrete offers the advantage of being a plastic material in the

sense that it is poured like fluid in a mould before it hardens, allowing a variety of shapes for beams and piers that could not be achieved in a simple way with steel materi-als and steel profiles Some arch bridges with relevant spans, using mass concrete, were made at beginning of the twentieth century, such as the Willeneuve – Sur‐Lot Bridge, in

1919, with an arch span of 97 m, by Eugéne Freyssinet François Hennebique was one of the first to understand the role of steel reinforcement in concrete to create a suitable bridge material In 1911, he ‘made’ (Hennebique was more a builder than a designer) the Risorgimento Bridge in Rome, achieving a 100 m span, but the development of rein-forced concrete bridges occurred with the work of the great Swiss engineer Robert Maillart who was the first to explore the potentialities of reinforced concrete to build magnificent arch bridges He explored the concept of relative stiffness between the arch

and the deck structure, introducing the concept of Maillart’s arch, an arch without

bending stiffness, achieved by reducing the stiffness in such a way that the arch follows

the pressure line working only under normal forces However, its thickness is enough to resist to arch buckling The arch works in this case as the opposite of a cable in a sus-pension bridge Among the bridges based on Maillart’s arch concept, the longest arch span, 43 m, was the one of the Tshiel Bridge built in 1925 Robert Maillart explored a variety of arch structures from built‐in arches to three‐hinged arches, such as those he adopted for his classical bridge, the Salginatobel Bridge, in Switzerland (Figure 1.28), built in 1930 with a 90 m span In 1901, Maillart was also the first to adopt a concrete box section for arches, in the Zuoz Bridge with a 30 m span structure At the beginning, concrete arch bridges were always associated with large wood scaffolding structures, some of them initially developed by R Coray, built by cantilever construction from each side until the closure at the mid‐section was reached One may even say the most rele-vant wood bridges in the world were the scaffolding of concrete arch bridges A detailed discussion on the historical development of scaffolding structures is presented in the excellent book by Troyano [4]

The concept of multiple arch bridges in reinforced concrete was adopted by Eugéne Freyssinet in 1930, in Plougastel Bridge This bridge, over the river Elorn in France, is made with three arches of a 186 m span each, following the concepts of previously built multiple arch metal bridges Many years before, in 1910, Freyssinet designed a beautiful multiple three arch bridge with very flat arches and a rise‐span ratio of only 1/14.7 The arches, each with a 77.5 m span, were initially built as three hinged arches but a few months after completion Freyssinet detected alarming increased deflections at the mid‐span sections due to creep effects in the concrete

He decided to recover the initial shape of the bridge by inserting jacks at the mid span hinge sections and to eliminate these hinges The beauty of the bridge was the result of the flatness of the arches but also due to the triangulated system adopted

to transfer the loads from the deck to the arches Unfortunately, this bridge was destroyed during the Second World War

The span lengths of concrete arch bridges were increasing, and one of the largest spans is still the 390 m long arch of the Krk Bridge (Figure 1.29), completed in 1979 in the former Yugoslavia

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Figure 1.28 The Salginatobel Bridge, 1930, Switzerland (Source: Photograph by Rama, https://

commons.wikimedia.org/wiki/File:Salginatobel_Bridge_mg_4077.jpg CC BY‐SA 2.0 FR).

Figure 1.29 The Krk reinforced concrete arch bridge, 1979, former Yugoslavia (Source: Photograph by

Zoran Knez/ https://commons.wikimedia.org).

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